Question

Suppose soft drink production at bottling plant is described by the production function:

? =??^?.?

where y is the number of cases of soft drinks produced and x is the number of hours of labor hired. We also assume the wage for labor is r, price for soft drink is p and fixed costs is $1,000.

  How many cases of soft drinks will be produced with 4 hours of labor?
  Find each of the following if 4 hours of labor are employed. Please include units.

Average product of labor:

Marginal product of labor:

Total variable cost:

Total fixed cost:

Total costs:

Average variable cost:

Average fixed cost:

Average total costs:

Answer 1

Given:

y = 5x^0.5

y is the number of cases of soft drinks produced

x is the number of hours of labor hired

The wage for labor is r

The price for a soft drink is p

fixed costs is $1,000.

For 4 hours of labor:

The number of cases of soft drinks produced:

Output produced at 4 hours of labor:

Putting x = 4

Therefore 10 cases of soft drinks are produced at 4 hours of labor.

The average product of labor:

The average product of labor (APL) is the output per unit of labor.

APL = Output/Labor units

Therefore here output = y = 5x^0.5

Labor units = 4 hours of labor

Therefore, the average product of labor is:

Putting x = 4 hours of labor

The average product of labor at 4 hours of labor = 2.5 cases of soft drinks.

Therefore the average product of labor at 4 hours of labor is 2.5 cases of soft drinks produced.

The marginal product of labor:

The marginal product of labor (MPL) is the additional output produced with an additional labor employed.

MPL is the derivative of the output function with respect to labor.

Now putting x = 4 hours of labor, we get the marginal product of labor at 4 hours of labor.

The marginal product of labor at 4 hours of labor = 1.25 cases of soft drinks

Therefore the marginal product of labor at 4 hours of labor is 1.25 cases of soft drinks produced.

Total variable cost:

The total variable cost (TVC) is the cost that varies with the output. here, the variable factor is labor. Therefore the total variable cost is the cost of labor.

here, labor hours are given to be 4

As the cost of per unit of labor is the wage of labor = r

Output produced = 10

Therefore Total variable cost at 4 hours of labor is $10r, where r is the wage rate.

Total fixed cost:

Total fixed cost (TFC) is the cost that doesn't vary with the output. So it is cost of the fixed factor. Here, as it is given that the cost of the fixed is $1000. Therefore total fixed cost will remain same for all the output and labor units.

Total fixed cost at 4 hours of labor is $1000

Total cost:

Total cost is the sum of total variable cost and the total fixed cost.

Therefore, Total cost at 4 hours of labor is:

Total cost = Total fixed cost + Total variable cost

Total cost = 1000+10r

Total cost = $(1000+10r)

Therefore the total cost at 4 hours of labor is $(1000+10r), where r is the wage rate.

Average variable cost:

Average variable cost (AVC) is the per unit variable cost, the variable factor is labor. So it is the per unit of labor cost.

Here, at 4 hours of labor

Total variable cost (TVC) = 10r

Output (y) at 4 hours of labor = 10 cases of soft drinks

Therefore,

Therefore the average variable cost at 4 hours of labor is $r.

Average fixed cost:

Average fixed cost is the per unit of the fixed cost.

here, at 4 hour of labor

Total fixed cost (TFC) = $1000

output (y) = 10 cases of soft drinks

Therefore, average fixed cost at 4 hours of labor is:

Therefore the average fixed cost at 4 hours of labor is $100.

Average total cost:

The average total cost (ATC) is the per-unit total cost. It is the sum of the average variable cost and average fixed cost.

ATC = AVC + AFC

Therefore at 4 hours of labor,

Average total cost = Average variable cost + Average fixed cost

Average total cost = r + 100

Therefore, the average fixed cost at 4 hours of labor = $(r+100), where r is the wage rate.